Tile instructional systems and methods of making and using same

ABSTRACT

A tile instructional system for presenting a linear growth problem and for allowing the linear growth problem to be solved comprises an input tray having at least a first tile and a second tile. The first tile has a first marking and the second tile has a second marking. Each of the first tile and the second tile is configured to selectively expand in accordance with its respective marking. The system includes a tile bed comprising a plurality of tile slots. Each of the plurality of tile slots is configured to receive a tile. The linear growth problem is solvable by situating in the tile bed at least one of the first tile and the second tile and by causing each tile situated in the tile bed to expand in accordance with its respective marking to fill the tile bed.

CROSS REFERENCE TO RELATED APPLICATIONS

This Application is a continuation of U.S. patent application Ser. No.16/137,151 filed Sep. 20, 2018 and titled “Liquid Flow InstructionalSystems and Methods of Making and Using Same”, which is acontinuation-in-part of U.S. patent application, Ser. No. 15/939,153filed Mar. 28, 2018 and titled “Method for efficiently teaching contentusing an adaptive engine and a physical input entry device”, which is acontinuation-in-part of U.S. patent application Ser. No. 15/369,699filed Dec. 5, 2016 and titled “Method for Efficiently Teaching ContentUsing an Adaptive Engine.” The '699 application is in-turn acontinuation-in-part of U.S. patent application Ser. No. 15/044,641filed Feb. 2, 2016, which is a continuation-in-part of U.S. patentapplication Ser. No. 14/833,033 filed Aug. 21, 2015, which is acontinuation-in-part of U.S. patent application Ser. No. 14/833,037filed Aug. 21, 2015. The '699 application also claims priority to U.S.Provisional Application, Ser. No. 62/116,707 filed Feb. 16, 2015, and toU.S. Provisional Application, Ser. No. 62/040,091 filed Aug. 21, 2014.The disclosure of each these applications is incorporated by referenceherein in its entirety.

BACKGROUND OF THE DISCLOSURE

Mathematics is a structured network of cognitive abstractions subject toprecise laws, originally presented almost entirely in prose text andnumerals. This approach was the norm until symbolic representation wasinvented around the 15th Century. The introduction of the symbolicrepresentation allowed people to understand and grasp the abstractnature of mathematics easier and quicker, though at a cost in requiringmastery of the notation and its precise grammatical rules. Resultantly,symbolic representation grew in popularity in mathematics and theassociated fields, eventually becoming the new norm and standard. Overthe years, symbolic representation became ingrained in mathematicproblems present in education, research, science, and engineering. Infact, symbol representation has been used for so long that people assumethat mathematic problems can be presented and solved only with symbolsand resultantly cannot discern the difference between the visualinterface, i.e. symbols, and mathematics itself. While extremelybeneficial for research and application purposes, symbolicrepresentation does hinder many people in understanding and usingmathematics. Numerous research studies going back to the early 1990shave shown that, when ordinary people are repeatedly presented withmathematical problems in a (non-symbolic) meaningful real-world orreal-world-like environment, they rapidly achieve a high level ofproficiency. This implies that the difficulties many people experiencein doing mathematics are primarily of a linguistic nature, also known asthe symbol barrier, and do not indicate a lack of mathematical thinkingcapacity.

Modern technology allows for new and novel means for representation ofideas and theories. The present disclosure relates in part to analternative representation for problems about linear growth functionsthat eliminates the traditional use of symbols to provide an alternativeand user-friendly interface for mathematics. More specifically, thepresent disclosure relates in part to a method of using a tiling system,which can be either a physical system or a simulated representationthereof, to visually represent and solve problems about linear growthfunctions, thus overcoming the symbol barrier. This alternative approachto representing mathematical problems may have significant potential,both for uses in mathematics and for educational purposes. The artisanunderstands that linear growth is a ubiquitous phenomenon in many walksof life, and ways to assist people in developing an understanding oflinear growth and to be able to reason productively about linear growthmay, accordingly, play a major role in mathematics education

FIELD OF THE DISCLOSURE

The disclosure relates generally to tile instructional systems andmethods. More specifically, the disclosure relates to using physical orother tile systems for providing instruction regarding linear growthfunctions.

SUMMARY

In an embodiment, a tile instructional system for presenting a lineargrowth problem and for allowing the linear growth problem to be solvedcomprises an input tray having at least a first tile and a second tile.The first tile has a first marking and the second tile has a secondmarking. Each of the first tile and the second tile is configured toselectively and linearly expand in accordance with its respectivemarking. The system includes a tile bed comprising a plurality of tileslots. Each of the plurality of tile slots is configured to receive atile. The system comprises a switch. The linear growth problem issolvable by situating in the tile bed at least one of the first tile andthe second tile and by causing each tile situated in the tile bed toexpand in accordance with its respective marking to fill the tile bed.The first marking comprises a left chevron marking and the secondmarking comprises a right chevron marking. The selective expansion ofeach tile situated in the tile bed is effectuated via the switch.

In another embodiment, a tile instructional system for presenting alinear growth problem and for allowing the linear growth problem to besolved comprises an input tray having at least a first tile and a secondtile. The first tile has a first marking and the second tile has asecond marking. Each of the first tile and the second tile is configuredto selectively expand in accordance with its respective marking. Thesystem includes a tile bed comprising a plurality of tile slots. Each ofthe plurality of tile slots is configured to receive a tile. The lineargrowth problem is solvable by situating in the tile bed at least one ofthe first tile and the second tile and by causing each tile situated inthe tile bed to expand in accordance with its respective marking to fillthe tile bed.

In yet another embodiment, a method for presenting and solving a lineargrowth problem comprises providing an input tray having at least firsttile and a second tile. The first tile has a first marking and thesecond tile has a second marking. The method includes configuring eachof the first tile and the second tile to selectively expand inaccordance with its respective marking. The method comprises providing atile bed having a plurality of tile slots, where each of the pluralityof tile slots is configured to receive a tile. The method includessituating in the tile bed at least one of the first tile and the secondtile. The method comprises solving the linear growth problem by causingeach tile situated in the tile bed to expand in accordance with itsrespective marking to fill the tile bed.

BRIEF DESCRIPTION OF THE SEVERAL VIEWS OF THE DRAWINGS

Illustrative embodiments of the present disclosure are described indetail below with reference to the attached drawing figures and wherein:

FIG. 1 is a schematic representation of a problem structure employed bya system (shown below in FIG. 13) for adaptively teaching content to auser, in an embodiment.

FIG. 2 is a flowchart depicting a high-level operation of the system ofFIG. 13, in an embodiment.

FIG. 3 is another flowchart depicting a high-level operation of thesystem of FIG. 13, in an embodiment.

FIG. 4 is a flowchart outlining a method used by the system of FIG. 13to determine whether the user's answering data matches a solution to aproblem presented to the user.

FIG. 5 is a flowchart outlining a method used by the system of FIG. 13to determine whether the user's answering data matches an optimalsolution to the problem.

FIG. 6 is a flowchart depicting a process used by the system of FIG. 13to designate a final iteration of the teaching process.

FIG. 7 is a flowchart depicting another process used by the system ofFIG. 13 to designate a final iteration of the teaching process.

FIG. 8 is a flowchart depicting another process used by the system ofFIG. 13.

FIG. 9 is a flowchart depicting a process used by the system of FIG. 13to make an initial assessment of the competencies of the user.

FIG. 10 is a flowchart depicting a process used by the system of FIG. 13to analyze performance scores of the user so as to place the user withina particular teaching topic from a series of teaching topics.

FIG. 11 is a flowchart depicting a process used by the system of FIG. 13to ensure adequate curriculum coverage for the user.

FIG. 12 is a flowchart depicting the workings of an entry module of thesystem of FIG. 13.

FIG. 13 is a schematic representation of the system for adaptivelyteaching content to the user, in an example embodiment.

FIG. 14 is a schematic representation of an example physical input entrydevice of the system of FIG. 13, here a gear system, in an embodiment.

FIG. 15 is a schematic representation of another example physical inputentry device of the system of FIG. 13, here a liquid flow instructionalsystem, in an embodiment.

FIGS. 16A-16C illustrate a puzzle being represented and solved using theliquid flow instructional system of FIG. 15.

FIGS. 17A-17F illustrate another puzzle being represented and solvedusing the liquid flow instructional system of FIG. 15.

FIGS. 18A-18C illustrate yet another puzzle being represented and solvedusing the liquid flow instructional system of FIG. 15.

FIG. 19 is a schematic representation of yet another example of aphysical input entry device of the system of FIG. 13, here a tilesinstructional system, in an embodiment.

FIGS. 20A-20D illustrates a linear growth problem being represented andsolved using the tiles instructional system of FIG. 19.

FIG. 21 is a table that illustrates example tiles and tile sections ofthe tile instructional system of FIG. 19 and growth rules associatedtherewith.

DETAILED DESCRIPTION

A major component of digitally implemented learning systems inmathematics (the field used in this application for illustrativepurposes) is the regular provision of problems or puzzles that need tobe solved to proceed. It is well established in mathematics educationthat to be most effective, problems or puzzles must be at the upperlimit of a user's ability at that moment—within what is known as theuser's zone of proximal development (ZPD). To achieve this aim, thesystem must constantly monitor the performance of the user to determine,dynamically, what the user's current ability level is, and to selectproblems or puzzles that keep the user in his or her ZPD. Sincemathematical problems or puzzles can be developed on a linear scale ofdifficulty, doing this is straightforward, and has been implemented onmany occasions in different systems. Use of such a linear scale ofdifficulty can work well in a system that focuses on one particularskill or technique. However, for a learning system that covers a rangeof topics, there is a tension between ensuring curriculum coverage andmaintaining the user in his or her ZPD.

Some embodiments of the present disclosure relate generally to the fieldof cognitive testing and adaptive learning. More specifically, someembodiments of the present disclosure include methods and systems foreffectively and efficiently teaching educational content using adaptivelearning and open-ended problems or puzzles. In embodiments, anindividual's performance is monitored while he or she is solving aproblem and the disclosed systems and methods utilize adaptive learningto select subsequent problems or puzzles of the requisite level ofdifficulty. This ensures that the individual is adequately challengedand is kept in his or her ZPD. At the same time, embodiments of thepresent disclosure ensure adequate coverage of each offered curriculumby requiring the individual to solve a specific problem from eachcurriculum; which if solved, demonstrates high degree of proficiency. Avariety of problems may be used for the present disclosure in order tosuit the education level for each individual. The problems may berepresented in the form of a puzzle or may be presented through avariety of mediums. The ideal problem, in embodiments, is an open-endedproblem that is presented to the individual in the form of a puzzle, agame essentially.

Some embodiments of the present disclosure also relate to systems usableto present such open-ended (or other instructional) problems, and tomethods of making and using such systems. These systems, in addition topresenting the instructional problems to the user, may be configured toallow the user to respond to the problems in a step by step fashion suchthat insight is gleaned into the user's problem solving thought process.Such insights into the user's thought process while solving problems mayallow the instruction to be better tailored to the user, as compared to,for example, the conventional multiple choice format used in schoolstoday. In an embodiment, the instructional system may be a gear system(such as a physical or virtual gear system). In another embodiment, andas discussed in more detail herein, the instructional system may be aliquid flow system (such as a physical or a virtual liquid flow system).In another embodiment still, the instructional system may be a tilessystem (such as a physical or a virtual tiles system). These differentinstructional systems may be generally configured to present to the user(and educate the user about, determine the user's mastery in, etc.)different types of problems. For example, the gears system may begenerally configured to present algebraic problems. The liquid flowsystem may be generally configured to present problems regardingproportional reasoning (e.g., fractional quantities, decimals,percentages, relative areas, etc.). The tiles system may be generallyconfigured to present problems relating to linear growth functions,e.g., simultaneous linear equations with a single unknown. And so on.The disclosure below discusses the various concepts outlined above inmore detail. Specifically, FIG. 1 and the associated discussion thereofin the disclosure relates to an organizational scheme for storingproblems when adaptively and efficiently teaching content to a user.FIG. 13 shows a system for adaptively and efficiently teaching contentto a user, and FIGS. 2-12 relate to various aspects thereof. FIG. 14relates to a physical or virtual gear system which may be a part of orwhich may be usable with the instructional system of FIG. 13. FIGS. 15,16A-16C, 17A-17F, and 18A-18C relate to a physical or virtual liquidflow system which may be a part of or which may be usable with theinstructional system of FIG. 13. FIGS. 19, 20A-20D, and 21 relate to aphysical or virtual tiles system which may be part of or which may beusable with the instructional system of FIG. 13. The disclosurediscusses these figures in-turn.

Referring to FIG. 1, the present disclosure includes a series ofteaching topics, wherein each teaching topic includes a lead problem anda plurality of secondary problems (Step A). Each of the teaching topicsis associated with a specific curriculum; a curriculum may be focused ona specific concept, puzzle type, theme, or a field of study. Forexample, one implementation of the present disclosure utilizes differentmathematical concepts and problem-solving challenges in order to make upthe series of teachings topics. The lead problem and secondary problemsfor each teaching topic all focus on the same curriculum. Each of theproblems is an open-ended problem or puzzle and can be solved in amultitude of ways, with each way being associated with an answer that issatisfactory according to a prescribed measure. More specifically, thelead problem and the secondary problems for each of the teaching topicsare associated with an optimal solution and at least one other solution(Step B). The optimal solution may be defined based on the least numberof steps used to solve the problem, the highest score attained insolving the problem according to a prescribed scoring system, the exactsequence of steps taken to solve the problem (“solution path”), and/orother similar characteristics. The other solution is any solution otherthan the optimal solution. The artisan will understand from thedisclosure herein that a problem may have two or more optimal solutions(e.g., where the optimal solution is defined based on the least numberof steps, two or more solutions may be deemed optimal where they eachinvolve the same (lowest) number of steps). Similarly, the artisan willappreciate that a problem may have two or more solutions other than theoptimal solution. Thus, each of the phrases “an optimal solution”, “theoptimal solution”, “other solution”, “the other solution”, etc., may butneed not refer to a solitary solution.

As can be seen in FIG. 1, in an embodiment of the present disclosure,the series of teachings topics is organized in a tree-like structure,comprising a central trunk and a multitude of branches. The centraltrunk comprises the lead problems for each of the teaching topicsarranged in a linear fashion. Each of the lead problems is furtherconnected to an emanating branch. Each branch comprises the secondaryproblems associated with the teaching topic of the lead problem. Thesecondary problems and lead problem for each of the teaching topics isfurther associated with a difficulty rank that is used to incrementallyincrease the problem difficulty for the individual. In an embodiment ofthe present disclosure, the difficulty rank of the lead problem isgreater than the difficulty rank of each secondary problem within eachof the teaching topics. Thus, the lead problem is used as a test for theassociated teaching topic. If the individual can effectively solve thelead problem for a specific teaching topic, then he or she may skip thesecondary problems of the specific teaching topic. This allows anindividual that has a high level of proficiency to quickly progressthrough the series of teaching topics to a curriculum that adequatelychallenges him or her without having to repeat content which he or shehas already mastered.

Embodiments of the present disclosure comprise a method and a system.The method delineates the rules and steps necessary to construct aspecific path for a user through the series of teaching topics. Thespecific path is based on the performance of the user and thus ismodified after each problem addressed by the user. The system comprisesthe physical components necessary to execute the method of the presentdisclosure. The system may comprise a personal computing (PC) device anda physical input entry device discussed further below. The PC deviceincludes a processor and a physical user interface (Step C). Asdiscussed herein, the physical user interface (or the physical inputentry device) may be a device not conventionally associated with genericcomputers. The processor executes the method of the present disclosurein the form of a software application at least in part. The computingdevice administers the series of teaching topics and the physical inputentry device allows the user to interact with the present disclosure tosolve and transition through the series of teaching topics. Type ofdevices that may be used as the PC device include, but are not limitedto, desktop computers, laptop computers, smartphones, tablets, and othersimilar electronic devices. Types of devices usable in the presentdisclosure as the physical input entry device are discussed furtherbelow. As discussed in more detail herein, in some embodiments, thefunctions of the physical input entry devices may be effectuated byvirtual devices, e.g., by means of interactive graphical user interfacesthat emulate these physical input entry devices and allow for thesolution path of the user to be captured and evaluated.

Two important aspects to note for the present disclosure: there are nomultiple-choice questions and the user must carry out all key steps ofthe problem or puzzle with the PC device. This allows the presentdisclosure to monitor and track every step that the user goes through(“solution path”) in order to solve the problem or puzzle, thusproviding raw descriptive information relating to the individual'scognitive/solving ability.

Referring to FIG. 2 and FIG. 3, the overall process of the presentdisclosure begins with the physical user interface prompting to solve aspecific problem within an arbitrary teaching topic, wherein thearbitrary teaching topic can be any topic within the series of teachingtopics (Step D). The user then attempts to solve the specific problemthrough the physical user interface. Answering data for the specificproblem is received with the PC device (Step E) to be analyzed. If theanswering data is not acceptable, then Steps D and E are repeated untilthe answering data matches either the optimal solution or the othersolution of the specific problem. Once a solution for the specificproblem is found, the user's performance is analyzed based on whichsolution was found and, resultantly, directed accordingly through theseries of teaching topics.

If the answering data of the specific problem matches the other solutionof the specific problem, then the user is directed to solve a nextproblem within the arbitrary teaching topic; the computing deviceprompts to solve the next problem within the arbitrary teaching topic(Step G). The other solution for the specific solution indicates averageproficiency in the curriculum of the arbitrary teaching topic. In whichcase, the user is directed to solve the secondary problems from thearbitrary teaching topic in order to practice, achieve mastery, andensure curriculum coverage before progressing to the next curriculum,i.e. the next teaching topic following the arbitrary teaching topic. Inother words, this conditional moves the user through the branch of thearbitrary teaching topic one problem at a time if any solution besidesthe optimal solution is entered. Alternatively, if the answering data ofthe specific problem matches the optimal solution of the specificproblem, then the user is prompted to solve the lead problem within anext teaching topic through the physical user interface (Step H). Thenext teaching topic is defined as the teaching topic following thearbitrary teaching topic within the series of teachings topics. Ingeneral, identifying the optimal solution for the specific problemsignifies that the user has the required degree of solution proficiencyfor the curriculum associated to the arbitrary teaching topic. Thus, theuser is permitted to skip the rest of the problems within the arbitraryteaching topic and jump to the next point in the trunk. This conditionendures that the user is kept within his or her ZPD at each step withinthe series of teaching topics.

Additionally, during Step H, if the specific problem is a last problemwithin the arbitrary teaching topic, then the user is prompted to solvethe lead problem within the next teaching topic, regardless whether theanswering data for the specific problem matches the optimal solution orthe other solution of the specific problem. Reaching and solving thelast problem within the arbitrary teaching topic indicates that the userhas reached an acceptable proficiency for the curriculum associated withthe arbitrary teaching topic and is thus permitted to move on to thenext teaching topic.

Finally, the last step in the overall process of the present disclosureis executing the aforementioned steps for the series of teaching topics.In particular, executing a first plurality of iterations for Steps Dthrough H with the processor by using either the next problem within thearbitrary teaching topic of an arbitrary iteration or the lead problemwithin the next teaching topic of the arbitrary iteration as thespecific problem of a subsequent iteration (Step I). This is executeduntil the arbitrary iteration is circumstantially designated as a lastiteration by the processor. The arbitrary iteration and the subsequentiteration are from the first plurality of iterations. Each of the firstiterations is Step D through H being executed for a particular problem;the particular problem is dependent on the user's real-time performanceand knowledge/proficiency of the curriculum being addressed.

The overall process of the present disclosure is executed until the userdemonstrates adequate proficiency in every teaching topic. In relationto the overall process, this is the case when the arbitrary iteration isdesignated as the last iteration. One such instance is when the usershows adequate proficiency in a final teaching topic by solving one ofthe problems from the final teaching topic with the optimal solution ofsaid problem; wherein the final teaching topic is the last topic withinthe series of teaching topics. Referring to FIG. 6, the user is finishedif the following conditions are met: (1) the teaching topic of thespecific problem is the final teaching topic; and (2) the answering datafor the specific problem matches the optimal solution of the specificproblem. If these conditions are met, then the arbitrary iteration isdesignated as the last iteration during Step H with the processor. Thus,indicating that the user has mastered the curriculum of the finalteaching topic and, resultantly, has finished the series of teachingtopics.

Another instance is when the user has reached and solved a last problemwithin the final teaching topic. Referring to FIG. 7, in relation to theoverall process, the user is finished if the following conditions aremet: (1) the teaching topic of the specific problem is the finalteaching topic; (2) the answering data for the specific problem matcheseither the optimal solution or the other solution of the specificproblem; and (3) the specific problem is the last problem within thefinal teaching topic. If these conditions are met, then the arbitraryiteration is designated as the last iteration during Step H by theprocessor, and the user finishes the series of teaching topics.

Referring to FIG. 4, if the user finds the other solution for thespecific problem, then he or she may be directed onto two differentpaths. The determining factor is if the specific problem is either thelead problem or one of the secondary problems. Prior to directing theuser, the processor first sequentially orders the secondary problemsrelative to the difficulty rank such that the user is incrementallyexposed to harder and harder problems. If the specific problem is thelead problem, then the user is directed to solve the secondary problemswithin the arbitrary teaching topic. More specifically, aleast-difficult secondary problem is chosen and designated as the nextproblem within the arbitrary teaching topic during Step G, wherein theleast-difficult secondary problem is from the plurality of secondaryproblems within the arbitrary teaching topic.

Alternatively, if the specific problem is one of the plurality ofsecondary problems, then the user is directed to solve the problem afterthe specific problem within the arbitrary teaching topic. In particular,a next-most-difficult secondary problem is designated as the nextproblem within the arbitrary teaching topic during Step G. Thenext-most-difficult secondary problem is from the plurality of secondaryproblems within the arbitrary teaching topic. Furthermore, it isimportant to note that the last problem referenced in Step H is thefinal problem within the arbitrary teaching topic. More specifically, amost-difficult secondary problem is designated as the last problemduring Step H; wherein the most-difficult secondary problem is from theplurality of secondary problems within the arbitrary teaching topic. Thefinal problem is the most difficult in order to test the user in thecurriculum of the arbitrary teaching topic.

Referring to FIG. 8, anytime during the overall process of the presentdisclosure the user is able to return to previously addressed problemsand attempt to find a different solution, in particular, the optimalsolution. In relation to the overall process, step C and step D may berepeated for a previous iteration during the arbitrary iteration, if thespecific problem from the previous iteration and the specific problemfrom the arbitrary iteration are within the arbitrary teaching topic,wherein the previous iteration is a designated number of iterations backfrom the arbitrary iteration. The designated number of iterations is setby an administrator account. Any problems further back than thedesignated number of iterations will not award the user with the abilityto skip to the next teaching topic if he or she identifies the optimalsolution. In alternative embodiments of the present disclosure, the usermay cross to previous topics in order to repeat problems. If the usermatches the answering data from the previous iteration to the optimalsolution for the specific problem from the previous iteration, then thesystem executes step H for the previous iteration during the arbitraryiteration.

Referring to FIG. 9, prior to allowing the user to solve the series ofteachings topics, the present disclosure first requires the user to passthrough an entry module. The entry module provides a rapid assessment ofthe user's ability and proficiency regarding the curriculums within theseries of teaching topics. The results from the entry module are used toplace the user within the series of teaching topics accordingly. Forexample, weak users are placed at an initial topic from the series ofteaching topics while stronger users may be allowed to skip a number ofearly topics.

The entry module includes a series of assessment problems, wherein eachassessment problem is associated with an optimal assessment solution andat least one other assessment solution, similar to the overall process(Step J). The series of assessment problems is populated with questions,problems, or puzzles of different curriculums, thus allowing the systemto fully determine the user's abilities. Additionally, the assessmentproblems may be easier than the problems from the series of teachingtopics. The process for the entry module is similar to the overallprocess of the present disclosure. First, the user is prompted to solvea specific assessment problem from the series of assessment problemsthrough the physical user interface (Step K). Next, the user solves thespecific assessment problem through the physical input entry device. Thesystem receives answering data for the specific assessment problem (StepL). Steps K and L are repeated until the answering data for the specificassessment problem matches either the optimal assessment solution or theother assessment solution of the specific assessment problem. The user'spath through the assessment problems is partially adaptive, i.e. thepath is dependent on the user's performance.

If the answering data matches the other assessment solution of thespecific assessment problem, then the user is incrementally moved to thenext problem within the series of assessment problems. In particular,the user is prompted to solve a first succeeding problem through thephysical user interface, wherein the first succeeding problem issequentially adjacent to the specific assessment problem along theseries of assessment problems (Step N). This is similar to the overallprocess.

If the answering data matches the optimal assessment solution of thespecific assessment problem, then the user is moved forward through theseries of assessment problems a pre-set number of steps. In particular,the user is prompted to solve a second succeeding problem through thephysical user interface, wherein the second succeeding problem issequentially offset from the specific assessment problem along theseries of assessment problems (Step 0). The offset, the number of steps,may vary depending on the specific assessment problem, the type ofeducational content, type of problems, or type of puzzles used for thepresent disclosure.

The user is maintained within the entry module until he or she reachesand solves a final problem within the series of assessment problems.More specifically, the processor executes a second plurality ofiterations for Steps K through O by using either the first succeedingproblem or the second succeeding problem of an arbitrary assessmentiteration as the specific assessment problem for a subsequent assessmentiteration. The second plurality of iterations is executed until thearbitrary assessment iteration is circumstantially designated as a lastassessment iteration by the processor. The arbitrary assessmentiteration and the subsequent assessment iteration are any sequentialpair of iterations within the second plurality of iterations.

Referring to FIG. 12, the present disclosure utilizes performance datafrom the entry module to determine where in the series of teachingtopics the user should be placed. In order to achieve this, performancecriteria are provided for each of the teaching topics. The performancecriteria quantify a minimum proficiency/ability necessary to solveproblems within the associated teaching topic. Once the user completesthe entry module, the processor assesses a performance score for each ofthe second plurality of iterations.

A variety of scoring methods may be used for determining the performancescore. Then, the performance score for each of the second plurality ofiterations is compiled into an overall performance score with theprocessor. The overall performance score is then compared to theperformance criteria for each teaching topic with the processor in orderto identify a set of matching topics from the series of teaching topics.The set of matching topics is the teaching topics within the series ofteaching topics which the user has shown proficiency in and thereforedoes not need to solve. This ensures that the problems addressed by theuser in the overall process of the present disclosure are within his orher ZPD.

Once identified, the set of matching topics is then displayed to theuser for selection. Referring to FIG. 10, the physical user interfaceprompts the user to select a specific topic from the set of matchingtopics. Once chosen, the selected topic is designated as the arbitraryteaching topic in Step D of an initial iteration from the firstplurality of iterations. This process assesses the user's ability andplaces him or her accordingly within the series of teaching topics.

In one embodiment, the present disclosure also includes a basics module,essentially a training area (also referred to herein as a tutor module).If at any point the system identifies that the user is struggling tosolve a problem, then he or she may be directed towards the basicsmodule. In one embodiment, certain problems within the entry module arededicated to separating users with strong and weak abilities. The basicsmodule tutors the user through basic elements utilized in the problemswithin the series of assessment problems and the series of teachingtopics. In order for the user to exit the basics module, the user mustcomplete all the problems and tasks within the basics module. Although,there is a one-time exit opportunity, if the user solves the firstpredetermined number of problems within the basics module by finding theoptimal solution in a single try for each one, then the user may exitthe basic module.

In an embodiment, a system for teaching content using an adaptive enginemay include one or more computing devices coupled to one or more inputentry devices (also referred to herein as an “interface device”). Theinput entry device coupled to the computing device(s) may be a physicaldevice other than a conventional computer component, such as a keyboard,mouse, a touchscreen display, etc. For example, in embodiments, theinput entry device may be a physical device that includes rotatablegears enmeshed with each other. Or, for instance, the physical inputentry device may comprise pieces of a puzzle that can be arranged inpredefined patterns. In these embodiments, the user may use the physicalinput entry device to solve one or more problems (e.g., puzzles or otherproblems) displayed elsewhere, e.g., on a display of the computingdevice. The computing device may evaluate the inputs provided by theuser via the physical input entry device and, based on this evaluation,adaptively select the next problem to be presented to the user. Asdiscussed above, and depending on the user input, the next problempresented to the user may be a problem within the same teaching topic ora different teaching topic (e.g., a lead problem of a different teachingtopic).

FIG. 13 shows an example system 100 for teaching content using anadaptive engine and a physical input entry device, as discussed herein.The system 100 may include a structure 102. The structure 102 may be acomputer, a server, a network of computer servers, etc., and is shownwith a processor 106 communicatively coupled to a network interface 108,an input/output device 109, and a memory 110. Processor 106 representsone or more digital processors. Network interface 108 may be implementedas one or both of a wired network interface and a wireless networkinterface, as is known in the art. The input/output device 109 mayinclude any suitable input/output device, such as a display, speakers, akeyboard, a mouse, a touchscreen, etc. Memory 110 represents one or moreof volatile memory (e.g., RAM) and non-volatile memory (e.g., ROM,FLASH, magnetic media, optical media, et cetera). Although shown withinstructure 102, memory 110 may be, at least in part, implemented asnetwork storage that is external to structure 102 and accessed vianetwork interface 108.

Software 114, a user database 116, and a problems database 117 may bestored within a transitory or non-transitory portion of the memory 110.Software 114 includes machine readable instructions that are executed byprocessor 106 to perform the functionality of structure 102 as describedherein. The user database 116 may include a plurality of records, eachpertaining to one of a plurality of users. For example, the userdatabase 116 may include a listing of lead problems attempted and/orsolved by each user, a listing of secondary problems attempted and/orsolved by each user, and other such user-specific information. The userdatabase 116 may, in embodiments, be omitted.

The problems database 117 may include a database of lead problems andassociated secondary problems, such as mathematical problems or puzzles,or other problems, arranged for example by teaching topic, concept type,puzzle type, theme, field of study, etc. The problems database 117 mayfurther include each or at least a plurality of solutions for eachproblem, including the optimal solution thereof, together with adifficulty rank for each problem. The software 102 may be configured topresent a user a lead problem, and subsequently, another lead problem ora secondary problem associated with the original lead problem, based onan input provided by the user via the input entry device (as discussedherein).

The online structure 102, using protocol 118 and Application ProgrammingInterface 132A, may communicate over a wired or wireless network 104with an input entry device 134 of a user 136. The user 136 may be anyindividual (or in embodiments, group of individuals) who are beingeducated and/or evaluated using the system 100 described herein.

Network 104, which is formed in part by one or more of the Internet,wireless networks, wired networks, local networks, and so on,facilitates communication between the structure 102 and the input entrydevice. The user 136 views a problem presented by the software 114 onthe input/output device 109, e.g., a display of or associated with theonline structure 102, and in response thereto, utilizes the input entrydevice 134 to solve the presented problem. The software 114 evaluatesthe input provided by the user 136 and, based on this evaluation,presents on the output device 109 another lead problem or a secondaryproblem having a different difficulty rank. The input entry device 134may include one or more sensors 134A to allow for relevant interactionof the user 136 with the components of the input entry device 134 to becommunicated to the software 114 (e.g., motion and/or rotation detectingsensors such as optical and/or magnetic sensors, pressure detectingsensors, temperature sensors, weight sensors, volume sensors, etc.). Inembodiments, the input entry device 134 may also include one or moreprocessors or other such devices to allow for the output of the sensors134A to be evaluated. In other embodiments, the input entry device 134may be devoid of a processor or other comparable device and the adaptiveengine 126 may be configured to decipher the output of the sensors 134A.In other embodiments, the input entry device 134 may be a stand-alonedevice (e.g., a battery operated or other dedicated device).

The input entry device 134A may further include, in addition to thesensing devices 134A, responsive devices 134B. The responsive devices134B may be configured to provide a controlled response in reaction tothe sensed input. The responsive devices 134B may be, for example, apump (e.g., a pump that causes liquid to flow from one location toanother based on a user input sensed by a sensor), a light or a speakerthat is activated when a puzzle is solved or during the puzzlepresentation, a cage that opens when a puzzle is solved by the user, abattery operated lever, a spring activated device, etc.

While the structure 102 is shown as having a solitary input entry device134 coupled thereto, in embodiments, the structure 102 may have amultitude of input entry devices 134 in communication therewith (e.g.,the structure 102 may be in communication with a statisticallysignificant number (such as hundreds of thousands) of input entrydevices 134). In these embodiments, each of the plurality of input entrydevices 134 may be associated with a unique user. The user, e.g., theuser 136, may also couple his or her input entry device 134 with thestructure 102 indirectly. For example, in embodiments, the structure 102may be an online structure (e.g., may be a webserver) and each user mayinteract therewith by coupling their respective input entry device 134to their personal (or other) computer which is in-turn coupled to thestructure 102. In embodiments, the system 100 may be a dedicated device(e.g., may be configured to effectuate only the purposes describedherein).

The software 114 may include an adaptive engine 126. The adaptive engine126 may include an evaluator 124. The adaptive engine 126 may initiallypresent to the user 136 a lead problem associated with a particulartopic via the input/output device 109. The user 136 may use the inputentry device 134 in an attempt to solve this lead problem. The user'sinput may be communicated to the structure 102 as answering data, andthe evaluator 124 thereof may evaluate the answering data to determineif the answering data includes or otherwise corresponds to the optimalsolution. If so, the evaluator 124 may subsequently present to the user136 via the input/output device 109 a suitable problem 127 which isassociated with a different teaching topic (see FIG. 1). Alternately, ifthe input provided by the user 136 via the input entry device 134includes a non-optimal solution, the evaluator 124 may present to theuser 136 the suitable problem 127 which, in this case, may be asecondary problem associated with the same teaching topic.

In embodiments, the software 114 may also include a performance module152, an entry module 154, and a tutor module 156. As is apparent fromthe disclosure herein, the adaptive engine 126, together with theperformance module 154, may monitor the user's performance 136 to ensurethat problems are presented to the user 136 so as to adequatelychallenge the user 136 while keeping the user 136 in his or her ZPD. Theentry module 154, also discussed above, may together with the adaptiveengine 126 initially present to the user 136 a series of assessmentproblems to allow the evaluator 124 to obtain a baseline assessment ofthe user's mastery over the teaching curriculum. The tutor module 156,also referred to as a basics module above, may be configured to tutorthe user 136, e.g., by teaching him or her about the basic elements of ateaching topic, based on a determination that the user 136 is strugglingto solve the presented problem.

As discussed above, the adaptive engine 126 may adaptively determine thesuitable problem 127 based on the input provided by the user 136 via theinput entry device 134. In embodiments, when determining the suitableproblem 127 to be presented to the user 136, the adaptive engine 126 mayalso take into account inputs provided by other users. For example,where inputs from a multitude of users indicate that a particularproblem within a teaching topic is easier to solve than the precedingproblem in that topic, the adaptive engine 126 may, based on theseinputs, adaptively change the difficulty rank of these problems in theproblems database 117. The artisan will understand that in so doing thesystem 100 may benefit from a statistically significant number of users136 (for instance, it may be more beneficial to adaptively change thedifficulty rank of a problem based on the input of many thousands ofusers as compared to changing the difficulty rank of a problem based onthe input of two or three users). Thus, use of a statisticallysignificant number of users may facilitate optimal operation of someembodiments of the system 100.

Workings of the disclosure will now be illustrated with an example. Theartisan will understand that the example is not intended to be limiting.

Focus is directed to FIG. 14 which shows an input entry device 200. Thisinput entry device 200 is but one example of the input entry device 134.The input entry device 200 is modeled after the gear system in U.S.patent application Ser. No. 14/833,037 filed Aug. 21, 2015, which, asnoted above, is incorporated by reference herein. The '037 applicationillustrates the workings of the physical gear system in detail, butdiscusses the physical gear system as a stand-alone device. A primarydifference between the physical gear system disclosed in the '037application and the physical gear system 200 is that the gear system 200is communicatively coupled to the structure 102, as illustrated in FIG.13 via the input entry device 134. The physical input entry device (orgear system) 200 is described herein to illustrate use of the system 100for teaching mathematical content, and particularly, algebraicequations, using the adaptive engine 126. The artisan will understandthat while mathematical content is used as an example to illustrate theworkings of the system 100, that the system 100 may likewise be used toadaptively teach other content to users (e.g., the user 136). Thedisclosure below first details the example input entry device 200, andthen outlines an example use of the input entry device 200 in the system100 to teach content to the user 136 adaptively.

The physical gear system 200 visually represents each entity of analgebraic equation and allows the user 136 to manipulate said entitiesthrough the individual gears of the gear system in order to determine asolution to the algebraic equation. Entities of the algebraic equationinclude a plurality of terms and at least one numerical constant,wherein one side of the equation is the plurality of terms and the otherside of the algebraic equation is at least one numerical constant. Eachof the plurality of terms includes a coefficient and a variable. Thevariable is a symbol that represents an undefined value within thealgebraic equation, while the coefficient is a constant number whichmultiples or amplifies the variable. Solving the algebraic equationincludes identifying a value for each of the variables, which wouldbalance the two sides of the algebraic equation.

The illustrated input entry device 200 includes a primary cog 1, aplurality of secondary cogs 2, and a fixed pointer 3. The primary cogrepresents a range of solutions for the algebraic equation and includesa plurality of teeth that is quantitatively greater than the numericalconstant. For example if the numerical constant is 20, than the numberof teeth on the primary cog would need to be greater than 20. Theplurality of teeth for the primary cog includes an origin tooth 4 and atarget tooth 5, each marked accordingly.

The origin tooth marks a starting point that the user 136 may referencein order to identify the remaining teeth within the plurality of teeth,essentially representing the zero value. The target tooth represents thenumerical constant of the algebraic equation. The target tooth isquantitatively offset from the origin tooth by the numerical constant,thus visually displaying the numerical constant as a radial increment onthe primary cog. Additional teeth may be marked on the primary cog toindicate their respective offset quantity from the origin tooth. In oneembodiment, each tooth on the primary cog is marked with a respectiveoffset quantity from the origin tooth. Alternatively, every incrementaltooth may be marked.

The plurality of secondary cogs represents the side of the algebraicequation relating to the plurality of terms. Each of the plurality ofsecondary cogs is associated with a corresponding term from theplurality of terms. This relationship is conveyed to the user byquantitatively matching a plurality of teeth on each secondary cog tothe value of the coefficient of its corresponding term. For example, ifthe corresponding term is “4x”, then the secondary cog representing thisparticular term would have four teeth. Each of the secondary cogs may bemarked with a readable label that depicts the coefficient of thecorresponding term, in turn conveying to the user the number of teethpresent on said secondary cog. Each of the secondary cogs is designed tomesh with and engage the primary cog such that rotation of each of theplurality of secondary cogs is used to drive the rotation of the primarycog. This includes matching the size and type of the teeth used for eachof the plurality of secondary cogs to that of the primary cog; a varietyof types of teeth may be used for the primary cog and thus the secondarycogs. As discussed herein, because the number of teeth of each of thethree secondary cog 2 is disparate, a full rotation of each secondarycog 2 will cause the primary cog to move by different amounts.

The fixed pointer indicates the current output for the input entrydevice 200, wherein the output corresponds to the side of the algebraicequation associated with the numerical constant. Additionally, the fixedpointer is used to zero/reset the gear system prior to solving thealgebraic equation. The gear system 200 is zeroed by positioning theorigin tooth coincident with the fixed pointer. The fixed pointer ispreferably shaped similar to an arrowhead and is positioned adjacent tothe primary cog, oriented towards the center of the primary cog.

In general, the method for solving the algebraic equation involvesaligning the target tooth at the fixed pointer, thus setting the currentoutput of the primary cog to be the numerical constant. This isaccomplished by first identifying a current tooth at the fixed pointer,wherein the current tooth is any one of the plurality of teeth on theprimary cog. If the current tooth is not the origin tooth, then theprimary cog is rotated until the origin tooth is set at the fixedpointer, essentially calibrating or resetting the input entry device200. Once the device 200 is reset, a plurality of rotations with one ormore of the plurality of secondary cogs is then executed in order torotate the primary cog so that the target tooth aligns with the fixedpointer. This alignment between the target tooth and the fixed pointeryields a possible solution for the algebraic equation. The potentialsolution lies in the number of rotations executed for each of thesecondary cogs. For example, two rotations of the secondary cog that isassociated with the term “4x” translates to the variable “x” being two.Once the target tooth is aligned with the fixed pointer, then theplurality of rotations is quantitatively identified for each of thesecondary cogs as a potential solution for the variable of thecorresponding term. The rotation direction of each of the secondary cogsrepresents either an increase or decrease in value for the variable ofthe corresponding term. A clockwise rotation by the secondary cogrepresents a quantitative increment in the potential solution of thevariable for the corresponding term. Similarly, a counterclockwiserotation by the secondary cog represents a quantitative decrement in thepotential solution of the variable for the corresponding term. Forexample, rotating one of the secondary cogs three turns clockwise andtwo turns counterclockwise means the value for the variable of thecorresponding term is one.

Positioning the target tooth at the fixed pointer yields a solution forthe algebraic equation, wherein the solution includes a potentialsolution for each of the variables, for each of the terms. However, thissolution is only one of many possible solutions for the algebraicequation. The most optimal solution in this example is achieved byminimizing the collective rotations of the secondary cogs 2. The leastamount of rotations for each of the plurality of secondary cogsrepresents the most efficient and optimal solution for the algebraicequation.

The input entry device 200 may also be used to solve the algebraicequation for a plurality of numerical constants, which is also known asa system of equations. Solving for the numerical constants includesrepeating the aforementioned method a multitude of times. That is, eachof the iterations is executed in order to solve the algebraic equationwith a corresponding constant from the numerical constants as one sideof the algebraic equation. Similar to solving for one numericalconstant, an initial iteration from within the plurality of iterationsincludes identifying the origin tooth as the current tooth and beginningthe plurality of iterations from the origin tooth. An arbitraryiteration from the plurality of iterations is defined as any iterationother than the initial iteration, while the previous iteration isdefined as the iteration that is executed prior to the arbitraryiteration. Solving for the numerical constants requires identifying thetarget tooth of the previous iteration as the current tooth of thearbitrary iteration. Consequently, the primary cog is not zeroed beforeeach iteration. For example, once the target tooth of each numericalconstant has been aligned to the fixed pointer, then a solution isidentified for the algebraic equation. An optimal solution in thisexample is achieved when a plurality of collective rotations isminimized during the iterations. The plurality of collective rotationsis defined as the summation of the rotations executed by each of thesecondary cogs during each iteration.

When solving the algebraic equation for more than one numericalconstants (e.g. a system of equations), the input entry device 200allows for constraints in the manner that a user solves for potentialsolutions. The present disclosure provides a plurality of constrainingcategories, each of which is associated with a priority rank. Theconstraining categories are used to guide the steps taken by the user tosolve the algebraic equation with the present disclosure. Each numericalconstant is assigned to a designated category from the plurality ofconstraining categories. This allows the system 100 to constraint anexecution sequence for the plurality of iterations in accordance to thepriority rank of the corresponding constant, and the priority rank isderived from the designated category of the corresponding constant. Theexecution sequence for the plurality of iterations provides the userwith a guide to optimize the manner in which to solve for the potentialsolutions of the algebraic equation.

Essentially, the execution sequence prompts the user to align the fixedpointer to one category of target teeth before aligning the fixed pointto another category of target teeth. The plurality of constraintcategories places restrictions on the manner on how the presentdisclosure can be used to solve the algebraic equation, similar to how asystem of equations can be solved in multiple ways but is stillmathematically constrained. The algebraic equation may but need notcontain only whole numbers. Also, in some embodiments, a sequential turnlimit may be applied to each of the secondary cogs in order to indicatethe number of rotations by a secondary cog has exceeded the most optimalsolution by a significant amount. Consequently, the plurality ofrotations with each of the secondary cogs 2 may not exceed thesequential turn limit.

In the illustrated embodiment, the input entry device 200 is implementedin the form of a physical apparatus. The physical apparatus 200 includesa multitude of gears and a support structure 202. The primary cog andthe secondary cogs are expressed by the gears. The gears are rotatablymounted to the support structure 202, e.g., on rotatable spindlesprovided thereon as shown in FIG. 14, and are positioned as describedherein. The user 236 may rotate the secondary cogs 2 (individuallylabeled A, B, and C for illustration) in order to identify the solutionto the algebraic equation. That is, in this example, to find a solutionto an algebraic equation presented to the user 136, the user 136 mustphysically rotate the secondary cog(s) A, B, and/or C. And, eachrotation of each secondary cog may be a physical action that may berecorded by the structure 102 and evaluated thereby to determine thepros and cons of the solution chosen by the user 136. The user input maybe communicated over the network 104A to the structure 102. For example,if the user 136 rotates the secondary cog A once clockwise and thesecondary cog C twice counterclockwise, each of these inputs may becommunicated to the structure 102 and evaluated by the software 114 asdiscussed herein.

In an embodiment, the adaptive engine 126 may present the problem to theuser 136 via the input/output device 109 (e.g., a display). The user 136may attempt to solve the problem displayed on the display 109 byphysically rotating one or more secondary cogs 2 of the input entrydevice 200. The adaptive engine 126, e.g., the evaluator 124 thereof,may evaluate these inputs to determine whether the user 136 provided theoptimal solution to the problem. If so, the adaptive engine 126, usinge.g., the performance module 152, may present to the user 136 via theinput/output device 109 a suitable problem 127 from a different teachingtopic. Conversely, if the evaluator 124 evaluates the user input anddetermines that the solution provided by the user 136 is a solutionother than the optimal solution, the subsequent suitable problem 127presented to the user 136 may be from the same teaching topic. Thedifficulty rank of the problems presented to the user 136 may beincreased or decreased by engine 126 in line with the user input. And,as discussed above, the difficulty rank assigned to a particular problemmay further be adaptively modified based on the inputs received by astatistically significant number of users.

Additional detail is now provided to illustrate how the input entrydevice 200 may be used to solve a problem—in this case, an algebraicequation—presented to the user 136 by the adaptive engine 126 via theinput/output device 109.

As can be seen in FIG. 14, the secondary cog A of the example inputentry device 200 has three teeth. Secondary cog B has five teeth. Andsecondary cog C has seven teeth. The primary cog 1 has 40 teeth. Thetarget tooth 5 is seven teeth away from the origin tooth 4 (i.e.,counting clockwise from the origin). Based on the configuration of theprimary cog and the secondary cogs, FIG. 1 may be represented by thefollowing equation:3x+5y+7z=7  [[Eq.1]]where the 3 in 3x refers to the number of teeth in secondary cog A, the5 in 5y refers to the number of teeth in secondary cog B, the 7 in 7zrefers to the number of teeth in secondary cog C, and 7 at the righthand side of the equation refers to the position of the target tooth ofthe primary cog relative to the origin tooth. The variable x refers tothe number of rotations of cog A (clockwise is positive and counterclockwise is negative), as also discussed herein. The variable y refersto the number of rotations of cog B. And variable z refers to the numberof rotations of cog C. The goal in this example is to rotate the primarycog so that the target tooth lands beneath the marker 3.

The artisan will appreciate that equation 1 has numerous solutions. Andeach of these solutions helps provide insight into the problem solvingprowess of the user 136. For example, a student Sam can use the inputentry device 200 of FIG. 14 to solve Equation 1 as follows. Sam mayphysically rotate cog C clockwise once. If cog C is rotated once in theclockwise direction, because it has seven teeth that are enmeshed withthe primary cog 1, the primary cog will move seven teethcounterclockwise. This would leave the target tooth below the marker 3.In terms of the symbolic equation, since cog A is not rotated, the valueof x is zero. Similarly, since cog B is not rotated, the value of y iszero. And because cog C is rotated once, the value of z is 1. Thisprovides one way to solve Equation 1.

-   -   x=0, y=0, z=1; [[ Sam's approach]]        i.e., 3(0)+5(0)+7(1)=7.

But, Equation 1 can also be solved in other ways. For example, a studentShelly may rotate cog B clockwise two times, and then rotate cog Acounter-clockwise once. That too will result in the target tooth landingbeneath the marker 3. In terms of symbols:

-   -   x=−1, y=2, z=0; [[ Shelly's approach]]        i.e., 3(−1)+5(2)+7(0)=7.

Both the solutions above are correct. But, in this example and as notedabove, the optimal solution is achieved by minimizing the collectiverotation of the secondary cogs. Sam's solution above required one stepwhereas Shelly's solution required two. Therefore, if this data set werethe only data set available, the system 100 may determine that Sam ismore proficient at solving algebraic equations than Shelly. Therefore,if the suitable problem 127 to be presented to each of Sam and Shellywere an algebraic equation, the adaptive engine 126 may subsequentlypresent an algebraic problem to Sam whose difficulty rank is greaterthan the difficulty rank of the algebraic problem presented to Shelly.

Indeed, the steps that the user 136 takes with the input entry device200 (and other such input entry devices) may provide much insight intothe user's problem solving abilities with respect to the teaching topicto which the problem belongs. Consider FIG. 14 again, but now assumethat secondary cog C is omitted. As will become clear from thediscussion herein, the representative equation would then be:3x+5y=7  [[Eq. 2]]

Assume that Sam solves Equation 2 by rotating cog B clockwise two timesand cog A counter-clockwise once (i.e., x=−1 and y=2). This would be themost efficient solution to Equation 2. However, to solve Equation 2 inthis manner, Sam must know that 2×5=10. That is, if Sam solves Equation2 in the manner just described, the adaptive engine 126 may determinethat Sam understands at least the basics of multiplication operations.The system 100 may therefore chose as a suitable problem (i.e., theproblem subsequently presented to Sam) a more complex problem involvingmultiplication or a problem in a different teaching topic (e.g.,division).

Assume now that Sam solves Equation 2 a different way. For example,assume Sam solves Equation 2 by rotating cog B clockwise once, rotatingcog A counter-clockwise once, and then by rotating cog B clockwise onceagain. This particular solution indicates that Sam is not proficient atmultiplication because he used only addition and subtraction to solveEquation 2. In this case, the adaptive engine 126 may subsequentlypresent to Sam a different problem (e.g., a problem in which thecomplexity of the addition is increased or a problem in which thecomplexity of the multiplication is decreased). In this way, thus, thesystem 100 may continually evaluate the progress of the user 136 andpresent to him or her problems that challenge the abilities of the user136 while ensuring that the user 136 is within his ZPD.

In embodiments, the physical input entry device 200 may be configurableby the user 136. For instance, and with respect to the input entrydevice 200 described as an example herein, the user 136 may be allowedto add or subtract gears from the device 200 (e.g., the supportstructure 202 may allow for the user 136 to: rotatably couple additionalsecondary gear(s) to the primary gear; remove one or more secondarygears; add or remove one or more teeth from the primary gear and/or thesecondary gear; use a differently sized primary gear, etc.). Suchselective configurability of the physical input entry device 200 mayfurther increase the versatility of the system 100. Other input entrydevices (e.g., device 300, device 700, etc.) discussed herein maylikewise be selectively configurable.

The artisan will appreciate from the disclosure herein that the gearsystem 200 is but one example of the input entry device 134, and otherinput entry devices for use with the adaptive system 100 for teachingcontent are also contemplated. FIG. 15, for instance, shows anotherexample 300 of the input entry device 134. The input entry device 300may also be referred to herein as a liquid flow instructional device300.

The disclosure relating to the liquid flow instructional device 300includes a method for representing a proportions problem and a methodfor solving the proportions problem. The method for representing theproportions problem utilizes the liquid flow instructional device 300 toexpress the proportions problem in a non-traditional fashion. The methodfor solving the proportions problem defines the steps necessary todetermine a set of values that solves the proportions problem using theliquid flow system 300, essentially identifying a solution to theproportions problem. The liquid flow instructional system 300 physicallyand visually represents each entity of the proportions problem (orpurely visually in the case of a digital implementation as describedherein) and allows the user to manipulate said entities through anadjustable valve 306 of the liquid flow system 300 to determine asolution to the proportions problem. Entities of the proportions problemmay include a number, a plurality of numbers, a geometric shape(circular disk, rectangle, or other regular shape), etc.

In more detail, the liquid flow instructional device 300 may include asupport structure 301S onto which a plurality of tanks and/or othercontainers configured to retain fluid are situated (e.g., mounted). Theplurality of tanks may include one or more input tanks and a pluralityof output tanks (e.g., two, three, or four or more output tanks, etc.).For example, in the example illustrated in FIG. 15, the liquid flowinstructional device 300 includes an input tank 302 and output tanks304A, 304B, and 304N. Each of the input tanks 302 and the output tanks304A, 304B, and 304N may have a pre-determined capacity which may bedisplayed thereon or elsewhere. Each of the input tank 302 and theoutput tanks 304A, 304B, and 304N may be fluidly and selectively coupledto each other. For example, a pipe may connect the input tank 302 to thevalve 306, and a plurality of pipes may connect the valve 306 to theplurality of output tanks. Thus, the adjustable valve 306 mayselectively and fluidly couple the input tank 302 to the output tanks304A, 304B, and 304N.

A user controlled binary switch, e.g., the fill button 308communicatively coupled to the valve 306, may initiate on user commandselective flow of the liquid from the input tank to the plurality ofoutput tanks. The initial state of the input tank 302 may be full (i.e.,the input tank 302 is full when a puzzle is presented to the user (e.g.,user 136)). The initial state of the output tanks 304A, 304B, and 304Nwhen the puzzle launches may be empty. The initial position of theswitch 308 may be off. The user may selectively manipulate the amount ofliquid that flows from the input tank 302 to each of the output tanks304A, 304B, and 304N via the adjustable valve 306, and morespecifically, via movable arms (e.g., rotatable arms, slidable arms,etc.) thereof. When the arms of the valve 306 have been set by the useras discussed herein, the user may set the switch 308 to on, whereuponthe entire contents of the input tank 302 may flow first to the valve306, and then to the plurality of output tanks 304A-304N as determinedby the valve settings. Upon completion of the flow, the switch 308 mayautomatically re-set to off and the input tank may automatically refill.When the puzzle requires a multi-step solution, as discussed herein, theinitial state of the output tanks at the start of any step may be theend-state of the previous step.

In embodiments, the input tank 302 (and/or output tanks 304A, 304B, and304N) may be fluidly coupled to a water source, such as a faucet, awater body, etc. to allow various amounts of fluid to be filled indifferently sized input tanks 302. As discussed herein, the adjustablevalve 306 may allow the user to selectively apportion liquid from theinput tank 302 into the output tanks 304A, 304B, and 304N. Inembodiments, and as discussed below, the adjustable valve 306 may have aplurality of arms (e.g., up to four arms) which the user may use toselectively apportion the liquid from the input tank 302 into the two ormore output tanks 304A, 304B, and 304N. The liquid flow instructionaldevice 300 may comprise a fill button 308 or other activation means,which, when employed by the user 136, may initiate fluid flow from theinput tank 302 to the one or more output tanks 304A, 304B, and 304N inline with the adjustable valve 306 settings set by the user. The user'sobjective may be to use the adjustable valve 306 to apportion liquidfrom the input tank 302 to the output tanks 304A, 304B, and 304N so asto exactly fill each of the output tanks 304A, 304B, and 304N withoutspillage. In puzzles requiring multi-step solutions, as will becomeclear from the disclosure herein, the fill button 308 may have to beemployed two or more times by the user to solve the puzzle. Inembodiments, e.g., where the physical input entry device 300 is beingused, the input entry device 300 may include means (e.g., pump(s),siphons, gravity fed devices, etc. (i.e., sensing devices 134A and/orresponsive devices 134B (see FIG. 13))) to cause the liquid to flow fromthe input tank 302 into the output tanks 304A, 304B, and 304N in linewith the adjustable valve settings set by the user. In embodiments, theinput entry device 300 may be a modular device such that the size and/ornumber of input and output tanks may be varied to create new puzzles.The size of the input tank 302 may but need not be the same as the sizeof the output tanks 304A, 304B, and/or 304N, and the output tanks maylikewise have different sizes. In embodiments, the size of the outputtanks 304A, 304B, and 304N may be the same but indicia may be providedto indicate that a different amount of liquid is to be filled in eachoutput tank to solve the puzzle. In embodiments, a volume of the inputtank 302 may be one of: (a) equal to a collective volume of theplurality of output tanks 304A, 304B, and 304N; and (b) a multiple ofthe collective volume of the plurality of output tanks 304A, 304B, and304N. The phrase “collective volume” indicates the actual collectivevolume of the output tanks and/or the collective volume thereof asindicated thereon or elsewhere.

FIG. 16A shows an example liquid flow instructional device 400. Theliquid flow instructional device 400 is substantially similar to theliquid flow instructional device 300, except as specifically notedand/or shown, or as would be inherent. Further, those skilled in the artwill appreciate that the embodiment 300 (and thus the embodiment 400)may be modified in various ways, such as through incorporating all orpart of any of the various described embodiments, for example. Foruniformity and brevity, corresponding reference numbers may be used toindicate corresponding parts, though with any noted deviations (forexample, the input tank is designated 302 in FIGS. 15 and 402 in FIG.16A, adjustable valve is designated 306 in FIGS. 15 and 406 in FIG. 16A,etc.). The artisan will appreciate from the disclosure herein that theconfiguration of the liquid flow instructional device 400 is one of themany possible configurations of the liquid flow instructional device300. In embodiments, the liquid flow instructional device 400 is anexample of the liquid flow instructional device 300.

In more detail, FIG. 16A shows the liquid flow instructional device 400in an initial condition (i.e., presenting a puzzle to the user), FIG.16B shows the liquid flow instructional device 400 in an intermediatecondition (i.e., where the user has employed the adjustable valve 406 toselectively apportion the liquid from the input tank 402 to the outputtanks), and FIG. 16C shows the liquid flow instructional device 400 in afinal condition (i.e., after the user has interacted with the fillbutton 408 to cause the liquid to flow from the input tank 402 to theoutput tanks 404A and 404B in line with the settings of the adjustablevalve 406). The adjustable valve 406 may have arms 406A and 406B thatmay allow the user to selectively define valve regions (or valve faceareas) 420A and 420B on the valve face. The valve face region 420A, inthis example, may correspond to the output tank 404A and the valve faceregion 420B may correspond to the output tank 404B. This correspondencemay be indicated by color coding (e.g., all or part of each of the valveface region 420A and the output tank 404A associated therewith may bered in color) or other means (such as by numerical identification orusing another visible indicator). In this example, the input tank 402has 100 units (e.g., mL, quarts, cups, gallons, etc.) of liquid. Theoutput tanks 404A and 404B are currently empty, and the goal here is touse the arms 406A and 406B of the adjustable valve 406 to apportion theliquid from the input tank 402 such that 75 units thereof end up in theoutput tank 404A and 25 units thereof end up in the output tank 404B.Specifically, the goal is to selectively configure the valve regions420A and 420B so as to exactly fill the output tanks 404A and 404B in aminimum number of attempts. As shown, the output tanks 404A and 404Bhave a capacity of 75 units and 25 units, respectively.

The artisan will appreciate from the disclosure herein that the problemrepresented in FIG. 16A is a proportions problem. A typical proportionsproblem may be: given a number N, and numbers M₁, . . . , M_(k), findnumbers R₁, . . . , R_(k) such that R₁+ . . . +R_(k)=1 andR_(i)×N=M_(i), for each i. For example, if N=200, k=3, M₁=70, M₂=80, andM₃=50, then a successful solution is to take R₁=0.35, R₂=0.40, andR₃=0.25, since 0.35+0.40+0.25=1 and 0.35×200=70, 0.4×200=80, and0.25×200=50. In the example shown in FIG. 16A, N=100, k=2, M1=75 andM2=25, so a successful solution is to take R1=0.75 and R2=0.25. That is,the input tank 402 represents the given number N, the output tanks 404A,404B, . . . 404 _(k) represent the numbers M₁, . . . , M_(k), and R1 andR2 represent the relative size of the valve regions 420A and 420B. Theuser may rotate or otherwise move the moveable valve arms 406A and 406Bto divide the face of the valve 406 into regions 420A and 420B thatrepresent the solution numbers R₁, . . . , R_(k). In some instances, thenumbers R₁, . . . , R_(k) may be restricted to come from a specifiedcollection; for example, fractions in the set {⅕, ⅖, ⅗, ⅘}, decimals inthe set {0.20, 0.4, 0.6, 0.8}, etc.

FIG. 16B shows the liquid flow instructional device 400 after the userhas properly configured the valve regions 420A and 420B to exactly fillup the output tanks 404A and 404B. While not expressly shown in FIG.16B, and as discussed above, in embodiments the valve regions and thecorresponding output tanks may be color coded (e.g., the region 420A andthe output tanks 404A may each include a red or other identifier toindicate correspondence therebetween and the region 420B and the outputtank 404B may each include a blue or other identifier to indicatecorrespondence between the region 420B and output tank 404B). Inembodiments, indicia may be provided to indicate the ratio of the sizeof the region 420A to the size of the region 420B (e.g., in FIG. 16B,the fraction ¾ may be provided in region 420A and the fraction ¼ may beprovided in the region 420B to indicate that the regions 420A and 420Brespectively take up ¾^(th) and ¼^(th) of the total valve region).Alternately, the indicia may include percentages, decimals between zeroand one, and/or other such indicators to indicate the relationshipbetween the valve regions 420A and 420B and/or the size thereof. Indiciamay also be provided to indicate the ratio of the volume of the outputtank 404A to the volume of the output tank 404B.

Once the valve arms 406A and 406B are set up as shown in FIG. 16B, theuser may depress or otherwise interact with the fill button 408. Theflow button 408 may be communicatively coupled to the input tank 402and/or the valve 406 (e.g., over a network, via a mechanical connection,etc.), such that interacting with the fill button 408 may initiate flowfrom the input tank 402 to the output tanks 404A and 404B. Thus, whenthe fill button 408 is depressed or otherwise interacted with, theliquid to flow from the input tank 402 to the output tanks 404A and 404Bin line with the valve regions 420A and 420B set by the user. Forinstance, interacting with the fill button 408 may cause a pump or otherliquid flow means to cause liquid to be pushed from the input tank 402into the output tanks 404A and 404B.

FIG. 16C shows the liquid flow instructional device 400 in its finalcondition, after the valve regions 420A and 420B have been set by theuser and the fill button 408 has been depressed. As can be seen in FIG.16C, the liquid has flown from the input tank 402 to the output tanks404A and 404B such that the output tanks are exactly filled (the phrase“exactly filled”, as used herein, means that a tank is filled tocapacity without spillage). Had the user configured the valve regions420A and 420B differently, e.g., if the user had configured the valveregions such that the region 420A had an area equal to that of region420B, output tank 404B would have overflown once the fill button 408 wasdepressed. As noted, the input tank 402 may be configured to be refilledimmediately (or at another time) for the presentation of the nextpuzzle.

FIG. 17A shows an example liquid flow instructional device 500 toillustrate a more complex problem than that shown in FIG. 16A. Theliquid flow instructional device 500 is substantially similar to theliquid flow instructional device 300 and 400, except as specificallynoted and/or shown, or as would be inherent. Further, those skilled inthe art will appreciate that the embodiment 500 may be modified invarious ways, such as through incorporating all or part of any of thevarious described embodiments, for example. For uniformity and brevity,corresponding reference numbers may be used to indicate correspondingparts, though with any noted deviations (for example, the input tank isdesignated 302 in FIG. 15, 402 in FIGS. 16A-16C, and 502 in FIG. 17A;the adjustable valve is designated 306 in FIG. 15, 406 in FIGS. 16A-16C,and 506 in FIG. 17A, etc.). In use, and in line with the disclosureherein, the problem in FIG. 17A may be presented to the user after theuser has solved the comparatively easier problem in FIG. 16A.

FIG. 17A shows the liquid flow instructional device 500 in an initialcondition (i.e., presenting a puzzle to the user), FIG. 17B shows theliquid flow instructional device 500 in an intermediate condition (i.e.,where the user has employed the adjustable valve 506 to apportion theliquid from the input tank 502 to the output tanks), FIG. 17C shows theliquid flow instructional device 500 in another intermediate condition(i.e., after the user has interacted with the fill button 508 to causethe liquid to flow from the input tank 502 to the output tanks 504A and504B in line with the settings of the adjustable valve 506 but beforethe output tanks 504A and 504B are exactly filled, FIG. 17D shows theliquid flow instructional device 500 in another intermediate condition(i.e., after the input tank 502 is refilled), FIG. 17E shows the liquidflow instructional device 500 in yet another intermediate condition(i.e., after the valve areas 520A and 520B have been set by the user 136for a second time), and FIG. 17F shows the liquid flow instructionaldevice 500 in a final condition (i.e., after the fill button 508 hasbeen depressed for the second time to complete the puzzle). In thisexample, the valve region 520A corresponds to the output tank 504A andthe valve region 520B corresponds to the output tank 504B. The inputtank 502 has 90 units of liquid, and can be refilled. The output tanks504A and 504B are currently empty, and the goal here is to use the arms506A and 506B of the adjustable valve 506 to apportion the liquid fromthe input tank 502 to exactly fill the output tanks 504A and 504B in aminimum number of attempts. Each of the output tanks 504A and 504B has acapacity of 90 units.

In the event that there are no numbers R₁, . . . , R_(k) such R₁+ . . .+R_(k)=1 and R_(i)×N=M_(i), for each i, completion of the puzzle mayrequire two or more applications of the settings. This may be referredto as multi-step problem herein, which is solved by obtaining a seriesof partial solutions, all but the final solution being a partiallycomplete configuration. The artisan will appreciate from the disclosureherein that the problem disclosed in FIG. 17A is a multi-step(specifically, a two-step) problem (i.e., the input tank having 90 unitswill need to be refilled once after the 90 units therein are transferredto the output tanks, because the output tanks 504A and 504B collectivelyrequire 180 units). Puzzles may likewise take a minimum of three or moresteps to resolve.

FIG. 17B shows the first setting of the control valve 506, and FIG. 17Cshows the results once the fill button 508 is first depressed. FIG. 17Dshows the input tank 502 being refilled (which may, in embodiments,happen automatically), FIG. 17E shows the second setting of the controlvalve 506, and FIG. 17F shows the final result once the fill button 508is depressed by the user the second time. In this way, the user may filleach of two 90 unit output tanks with a refillable 90 unit input tank intwo tries (i.e., using the fill button 508 twice). In essence, FIGS.17A-17F show the following:

-   -   Turn 1: Valve areas 520A and 520B take up ⅔^(rd) and ⅓^(rd) of        the valve face, respectively; and    -   Turn 2: Valve areas 520A and 520B take up ⅓^(rd) and ⅔^(rd) of        the valve face, respectively.

The artisan will appreciate that the puzzle may likewise be solved inother ways, including in two (or a different number of) attempts. Forexample, the puzzle may he been solved as follows:

-   -   Turn 1: Valve areas 520A and 520B take up ½ and ½ of the valve        face; and    -   Turn 2: Valve areas 520A and 520B take up ½ and ½ of the valve        face.

Had the user taken more than two tries to solve this puzzle, the system100 may have gleaned that the user does not have mastery overproportions problems and may have presented to him additional (e.g.,easier) proportions puzzles to solve.

FIG. 18A shows an example liquid flow instructional device 600 toillustrate another problem, wherein the numbers R₁, . . . , R_(k) areindicated as percentages. The liquid flow instructional device 600 issubstantially similar to the liquid flow instructional device 300, 400,and 500, except as specifically noted and/or shown, or as would beinherent. Further, those skilled in the art will appreciate that theembodiment 600 may be modified in various ways, such as throughincorporating all or part of any of the various described embodiments,for example. For uniformity and brevity, corresponding reference numbersmay be used to indicate corresponding parts, though with any noteddeviations.

FIG. 18A shows the liquid flow instructional device 600 in an initialcondition (i.e., presenting a puzzle to the user), FIG. 18B shows theliquid flow instructional device 600 in an intermediate condition (i.e.,where the user has employed the adjustable valve 606 to apportion theliquid from the input tank 602 to the output tanks 604A, 604B, and604C), and FIG. 17C shows the liquid flow instructional device 600 in afinal condition (i.e., after the user has interacted with the fillbutton 608 to cause the liquid to flow from the input tank 602 to theoutput tanks 604A, 604B, and 604C in line with the settings of theadjustable valve 606). In this example, the valve region 620Acorresponds to the output tank 604A, the valve region 620B correspondsto the output tank 604B, and the valve region 620C corresponds to theoutput tank 604C. The input tank 602 has 200 units of liquid, and can berefilled. The output tanks 604A, 604B, and 604C are currently empty, andthe goal here is to use the arms 606A, 606B, and 606C of the adjustablevalve 606 to apportion the liquid from the input tank 602 to exactlyfill the output tanks 604A, 604B, and 604C in a minimum number ofattempts. The output tanks 604A, 604B, and 604C have a capacity of 70units, 80 units, and 50 units, respectively.

FIG. 18B shows the device 600 in the intermediate condition, after theregions 620A, 620A, and 620C have been set to cause the output tanks604A, 604B, and 604C to be filled exactly. FIG. 18C shows the finalresult once the fill button 608 is depressed by the user. While in thisexample the numbers R₁, . . . , R_(k) are expressed as percentages, theartisan will appreciate that these numbers may likewise be expressed asfractions, decimals, or without any symbols (e.g., the user may simplyevaluate the valve 606 to discern the relative size of the valveregions). While not expressly shown in the figures, the correspondencebetween the valve regions 620A, 620B, and 620C to the output tanks 604A,604B, and 604C, respectively, may be indicated by color coding, numericindicia, or other appropriate means.

Proportions problems of the general nature discussed herein may providea proven and effective way to develop an understanding of, and afacility to manipulate, fractions, and proportions in a variety ofsettings. Thus, by playing such games, e.g., the ones shown in FIGS.16A-16C and 17A-17F, and 18A-18C, the user may get a deeperunderstanding of fractions and proportions. Applicant's research hasconfirmed that this is indeed the case. The particular proportionsproblems may be formulated specifically to facilitate the constructionof a physical or digital device that provides a learning experience thatbreaks the symbol barrier. In some embodiments, instead of exactlyfilling the output tanks, the puzzle may be solved by spilling the oneor more output tanks by a particular amount (or by no more than aspecified amount). Such may help develop the understanding ofproportions to include estimation skills.

While FIGS. 16A-16C, 17A-17F and 18A-18B show that the adjustable valveis generally circular, such is merely exemplary, and the valve (and theregions thereof formed by the arms) may be rectangular, square, or takeon other regular or irregular shapes. Similarly, the valve arms mayextend radially as shown (akin to the hands of a clock), but may extendvertically, horizontally, or in other directions in other embodiments.

FIG. 19 shows yet another embodiment 700 of the input entry device 134.The input entry device 700 may, like the other input entry devices 134disclosed herein, be a physical device that is communicatively coupledto the structure 102. Alternately, the input entry device 700 (and theother input entry devices) may be implemented digitally, e.g., via agraphical user interface and machine readable instructions. The inputentry device 700 may be directed to presenting and solving simultaneouslinear equations in a single unknown, with the intention of developing adeep and productive understanding of linear growth functions. The inputentry device 700 may also provide exercise in spatial reasoning. Theartisan will understand from the disclosure herein that linear growth isa ubiquitous phenomenon in many walks of life, and that assisting peoplein developing an understanding of linear growth can accordingly play amajor role in mathematics education.

The disclosure relating to the input entry device 700 includes a methodfor representing and solving a problem involving a linear growthfunction and simultaneous linear equations in a single unknown. Theprimary objective of the input entry device 700 is not to demonstrate tousers how systems of linear equations may be solved by hand. Rather, inembodiments, a primary objective of the input entry device 700 may be tocultivate in users a meaningful understanding of linear growth and toallow them to reason successfully about linear growth situations. Lineargrowth lends itself to instantiation in a simple mechanical device. Thedisclosure incorporates an element of engaging, challenging spatialreasoning that may provide a visualization of the growth.

Linear equations may normally be written in the symbolic algebraic form:y=ax+b, where x is an input variable, y an output variable, and a, b areconstants. They can be viewed both statically and dynamically.

Viewed statically, linear equations may capture a specific relationbetween two numbers. For example, for the equation y=3x+4, the equationsays what when x=7, then y=25, so the equation outlines a relationshipbetween 7 and 25. In other words, the equation specifies an algorithmthat, given a number x, produces a number y.

Viewed dynamically, a linear equation may specify a function. One commonway to represent the function defined by the equation y=ax+b is bydrawing its graph. While effective, the graphical representation mayobscure the inherently dynamic, procedural aspect of a function. Theinput entry device 700 may provide an alternative representation of sucha function that brings out the dynamic feature, drawing the user'sattention to the growth-pattern of the function.

Focus on the growth pattern of linear functions may be achieved byrepresenting the function in terms of small tiles or blocks that havepre-specified linear growth patterns. Of course, the tiles may take onother regular or irregular shapes. The input entry device 700 may alsobe referred to herein as a tile or block instructional system 700.

In more detail, the tile instructional system 700 may include a supportstructure 701S onto which one or more remaining components of the device700 may be situated. The device 700 may include an input tray 702. Theinput tray 702 may comprise one or more individual movable tiles (orblocks) 704 and/or movable tile sections 706. Each tile section 706 maycomprise a plurality of individual tiles 704 that are grouped together.

One or more of the tiles 704 (i.e., one or more of the individual tiles704 and/or one or more of the tiles 704 forming the tile sections 706)may include a growth rule. The growth rule may be indicated by, e.g.,Chevron markings or other indicia. For example, as shown in FIG. 19, atile may have a left chevron marking 708 or a right chevron marking 710.The term “chevron marking”, as used herein, includes any marking thatindicates a direction, such as a left arrow, a right arrowhead, etc. Thechevron markings 708, 710, when provided on tile sections 706, may beprovided on the leftmost and/or the rightmost tile 704 of that section706. As shown in FIG. 19, some tiles 704 may be devoid of any growthfunctions (indicated here by chevron markings) and other tiles 704 mayinclude a plurality of chevron markings.

Each chevron marking may indicate a growth rule. For example, a solitaryleft chevron marking 708 on a tile 704 may indicate that the particulartile 704 can grow a tile to the left. For example, when the growthfunction is invoked on a solitary tile 704 having a left chevronmarking, that tile 704 may grow a tile to the left and become a two tilesection. A solitary right chevron marking 710 on a tile 704 may indicatethat the particular tile 704 can grow a tile to the right. For example,when the growth function is invoked on a solitary tile 704 having aright chevron marking, that tile 704 may grow a tile to the right andbecome a two tile section.

As shown in FIG. 19, some tiles sections 706 may have each of a leftchevron marking and a right chevron marking; for example, the right mosttile of a tile section 706 may have a right chevron marking, and theleft most tile of a tile section 706 may have a left chevron marking.When the growth function is invoked, the right most tile may grow a tileto its right and the left most tile may grow a tile to its left. Thatis, on initiating a move (e.g., a Grow move), each tile 704 with achevron marking may expand in the direction of the chevron and theexpansion thereof may correspond to the number of chevrons associatedtherewith. For example, the particular tile section 712 which is shownas having four tiles 704 may, upon invocation of the grow move,initially expand one unit square to the left and two unit squares to theright to form a tile section having an overall length of 7 unit squares,with a single left chevron pointed at its leftmost end and a doubleright-pointed chevron at its right most end. This expanded tile, onceexpanded, may be expanded yet again via a second Grow move that willcause this tile section to have an overall length of 10 unit squares.And so on. A tile having no chevron marking, conversely, may not grow asit does not have growth markings thereon.

Invocation of the growth function may be effectuated in the physicaldomain (e.g., mechanically) or virtually. For example, in physicalembodiments, each tile with a chevron marking may have one or moreadditional tiles stacked atop the lowermost tile, and one or more ofthese tiles may be configured to extend outward (i.e., to the left or tothe right depending on the chevron marking) when the Grow move (orgrowth function) is invoked. In one embodiment, one or more of the tilesstacked above the lowermost tile may be spring loaded and may beconfigured to extend to the left or the right by the force of a springwhen the growth function is invoked. In another embodiment, powered(e.g., battery operated) arms may be used to cause a tile to extend tothe left or the right of another tile. In another embodiment still, thetiles may be nested (akin to Russian dolls) within each other and may bepushed out using mechanical means when the growth function is invoked.

The tile instructional system 700 may further include one or more tilebeds, such as tile beds 720A, 720B. Each tile bed 720A and 720B mayinclude a label indicating the total number of unit tile receiving slotsin that bed (e.g., 18 and 4 in beds 720A and 720B, respectively). Inanother embodiment, each tile receiving slot in each tile bed may benumbered to assist the user 136 in the determination of a solution.

The goal of the user 136 solving the linear growth problem presented bythe device 700 may be to take the appropriate tile(s) and/or tile bedsfrom the input tray 702 and position them in the tile beds 720A and 720Bin such a way that, by invoking a minimum number of Grow moves, all tilebeds are filled exactly (with no overlapping). Activation of a Grow movemay cause all tiles that have been placed in a bed to grow according totheir specified growth rule. There may be any number of trays and/ortile beds. The device 700 may provide a mechanism for solvingsimultaneous linear equations, with a focus on mathematical growth rules(functions), as discussed herein. In this example, the optimal solutionmay be of two forms: fewest number of applications of the Grow move andfewest number of tiles used. If the user 136 determines an optimalsolution to a problem, a different (more complex problem from the sameor a different topic) may be presented to the user 136. Alternately, ifthe user 136 is unable to determine the optimal solution, a different(e.g., a less complex problem from the same topic) may be presented tothe user 136. In this way, thus, the input entry device 134 may allowthe system 100 to educate the user 136 while ensuring that the user 136remains within his or her ZPD.

In embodiments, the device 700 may include an activator or binaryswitch, such as a grow button 722, which, when activated, may cause thetiles and tile sections to expand in accordance with their respectivegrowth functions. In physical embodiments, the grow button 722 may becommunicatively coupled to the tile sections (e.g., using RF or othernetwork). In other embodiments, the growth function of a tile and/ortile section may be activated by interacting with (e.g., depressing) thechevron marking thereon. In some embodiments, a deactivator (such as an‘ungrow’ or undo button 724) may be provided to reverse a grow move. Thetile instructional system 700 may, in embodiments, be modular to allowfor different puzzles to be presented to the user 136 (e.g., a tile bedmay be replaced with a differently sized tile bed, a tile section may bereplaced with a differently sized and/or marked tile section, etc.). Asdiscussed above, the puzzles may increase or decrease in complexitydepending on user progress. In embodiments, a guide having a pluralityof linear growth problems of various difficulties may be provided sothat the user 136 and/or another (e.g., an educator) may configure thedevice 700 to present and solve different problems. A similar guide maylikewise be provided in association with the other input devicesdiscussed herein.

FIG. 20A shows an example tile instructional system 800 for educatingusers about linear growth functions. The tile instructional system 800is substantially similar to the tile instructional system 700, except asspecifically noted and/or shown, or as would be inherent. Further, thoseskilled in the art will appreciate that the embodiment 800 (and thus theembodiment 700) may be modified in various ways, such as throughincorporating all or part of any of the various described embodiments,for example. For uniformity and brevity, corresponding reference numbersmay be used to indicate corresponding parts, though with any noteddeviations (for example, the input tray is designated 702 in FIGS. 19and 802 in FIG. 20A). The artisan will appreciate from the disclosureherein that the configuration of the tile instructional system 800 isone of the many possible configurations of the tile instructional system700. In embodiments, the tile instructional system 800 is an example ofthe tile instructional system 700.

As can be seen, the device 800 has in its input tray 802 three tilesections 806A, 806B, and 806C each comprising individual tiles 804. Inthis example, each of the tile sections 806A, 806B, and 806C have twoindividual tiles 804. In the tile section 806A, one tile has a leftchevron marking and the other tile has a right chevron marking. In thetile section 806B, the left tile has a left chevron marking and theright tile has two right chevron markings. In the tile section 806C, theleft tile has two left chevron markings and the right tile has two rightchevron markings. As discussed above, when the Grow move is initiated(e.g., using the grow button 822), each tile may expand in accordancewith the chevron markings thereon. The tile instructional system 800further includes two tile beds 820A and 820B. As shown, each tile ofeach tile bed may be numbered, though such is not required in allembodiments. The tile bed 820A has 6 tiles and the tile bed 820B has 10tiles. Of course, a different number and configuration of tiles and/ortile sections may be provided in the input tray and/or a differentnumber and configuration of tile beds may likewise be provided.

It will be appreciated that FIG. 20A shows the tile instructional device800 in an initial condition (i.e., presenting a linear growth problem tothe user 136). FIG. 20B shows the tile instructional device 800 in anintermediate condition (i.e., where the user has placed at least one ofthe tile sections 806A, 806B, and 806C into each of the tile beds 820Aand 820B). Specifically, as can be seen, the user 136 has placed thetile section 806A in the tile bed 820A such that it covers the third andthe fourth tile receiving slot of the tile bed 820A. The user 136 hasfurther placed the tile section 806C in the tile bed 820B such that itcovers the fifth and the sixth tile receiving slot of the tile bed 820B.

FIG. 20C shows the tile instructional device 800 in another intermediatecondition (i.e., after the user has activated one Grow move). As can beseen, in response to the Grow move, the tile section 806A has grown fromtwo tiles to four tiles, and more specifically, the tile section 806Ahas expanded one tile to the left and one tile to the right such that itnow covers slots 2, 3, 4, and 5 of the bed 820A. The chevron markingscontinue to be on the outermost tiles, but the left chevron marking isnow on the tile of the tile section 806A that corresponds to slot number2 of the bed 820A, and the right chevron marking is now on the tile ofthe tile section 806A that corresponds to slot number 5 of the bed 820A.The tile bed 820B has likewise expanded in response to the activation ofthe Grow move. However, as can be seen, because originally each of thetwo tiles of the tile section 806C has two chevron markings each, thetile section 806C has expanded from two tiles to six tiles.

FIG. 20D shows the tile instructional device 800 in the final condition(i.e., after the user has initiated the Grow move a second time to filleach tile bed 820A and 820B exactly). Once the user initiates anotherGrow move, the tile section 806A expands from four tiles (as shown inFIG. 20A) to six tiles to fill the tile bed 820A. Similarly, once theuser initiates another Grow move, the tile section 806B expands from sixtiles (as shown in FIG. 20A) to ten tiles to fill the tile bed 820B.

The artisan will understand from the disclosure herein that the tilesare representations of linear equations. Unlike the gears systemdisclosed above, where the focus is on solving equations in severalvariables, with tiles the focus is on linear functions as a way todescribe growth processes. An element of spatial reasoning is alsoincorporated in the positioning of the tiles and tile sections. FIG. 21shows some example growth rules, i.e., example tiles and tile sectionsand linear functions corresponding thereto. The examples in FIG. 21 arenot exhaustive and are not meant to be independently limiting.

The artisan will understand that while the disclosure focuses on aphysical input entry device 134 usable by the user 136 to provide inputsthat are then captured and evaluated by the structure 102, that in otherembodiments, the input entry device 134 may be provided as software withwhich the user 136 may interact via conventional means (e.g., via akeyboard and mouse, etc.). For example, an interactive graphical userinterface may comprise the gears system shown in FIG. 14, the liquidflow system shown in FIGS. 15, 16A-16C, 17A-17F and 18A-18B, the tilessystem shown in FIGS. 19 and 20A-20C, etc., and the user may interactwith these systems using conventional computer means (e.g., via akeyboard, a mouse, or other controller). However, in some embodiments,it may be preferable to use the physical input entry device 134 at leastbecause the real-world experience provided thereby may be more memorablefor the user 136 (as compared to pressing the keys of a keyboard and/ormoving the mouse to cause virtual gears on the screen to rotate in likefashion).

Many different arrangements of the various components depicted, as wellas components not shown, are possible without departing from the spiritand scope of the present disclosure. Embodiments of the presentdisclosure have been described with the intent to be illustrative ratherthan restrictive. Alternative embodiments will become apparent to thoseskilled in the art that do not depart from its scope. A skilled artisanmay develop alternative means of implementing the aforementionedimprovements without departing from the scope of the present disclosure.

It will be understood that certain features and subcombinations are ofutility and may be employed without reference to other features andsubcombinations and are contemplated within the scope of the claims. Notall steps described herein and/or listed in the various figures need becarried out or need to be carried out in the specific order described.

The disclosure claimed is:
 1. A computer-implemented method ofrepresenting a mathematical problem configured to educate a user onlinear growth functions, said method comprising: providing a graphicaluser interface for displaying an instructional system, saidinstructional system comprising: an input area comprising a tilesection, said tile section initially comprising at least two tiles andhaving associated therewith a growth rule, a number of tiles in saidtile section configured to increase in accordance with said growth rule;an output area having a predefined number of tile receiving regions,said tile section being selectively movable by said user to a locationat said output area; and a switch for causing said tile section to growin accordance with said growth rule after said tile section has beenmoved to said location; receiving said tile section from said input areato said output area based on user selection of said location; andsimulating growth of said tile section in accordance with said growthrule upon activation of said switch by said user; wherein, saidmathematical problem is solved when a final number of tiles in said tilesection equals said predefined number of tile receiving regions.
 2. Themethod of claim 1, wherein said tile section comprises a plurality oftile sections.
 3. The method of claim 2, wherein said user is requiredto select an appropriate tile section from said plurality of tilesections.
 4. The method of claim 1, wherein said switch is activated bysaid user a plurality of times to solve said mathematical problem. 5.The method of claim 1, wherein each of said at least two tiles in saidtile section grows a different number of tiles upon activation of saidswitch.
 6. The method of claim 1, wherein said growth rule represents alinear equation.
 7. The method of claim 1, wherein said output areacomprises a plurality of output trays, each of said plurality of outputtrays having a predefined number of tile receiving regions.
 8. Themethod of claim 1, further comprising representing a new mathematicalproblem based on an evaluation of user inputs.
 9. A computer-implementedmethod of representing a mathematical problem, comprising: providing agraphical user interface for displaying an instructional system, saidinstructional system comprising: an input area comprising a tilesection, said tile section having a growth rule associated therewith; anoutput area having a predefined number of tile receiving regions; and aswitch for causing said tile section to grow in accordance with saidgrowth rule; receiving said tile section from said input area to saidoutput area based on an input from a user; simulating growth of saidtile section based on said growth rule upon activation of said switch;wherein, said mathematical problem is solved when a final number oftiles in said tile section equals said predefined number of tilereceiving regions.
 10. The method of claim 9, wherein said growthpattern is indicated on said tile section by chevron markings.
 11. Themethod of claim 9, further comprising providing a plurality of outputtrays in said output area.
 12. The method of claim 9, wherein said tilesection comprises a plurality of tile sections.
 13. The method of claim12, wherein said user is required to select an appropriate tile sectionfrom said plurality of tile sections.
 14. The method of claim 9, whereinsaid switch is activated by said user a plurality of times to solve saidmathematical problem.
 15. The method of claim 9, wherein said growthrule represents a linear equation.
 16. The method of claim 15, whereinsaid linear equation contains a single unknown.
 17. Acomputer-implemented method of representing a mathematical problem,comprising: providing a graphical user interface for displaying aninstructional system, said instructional system comprising: an inputarea comprising a tile section, said tile section having a growth ruleassociated therewith; an output area having a predefined number of tilereceiving regions; a switch for causing said tile section to grow inaccordance with said growth rule; receiving said tile section from saidinput area to said output area based on an input from a user; simulatinggrowth of said tile section based on said growth rule upon activation ofsaid switch; wherein, said mathematical problem is solved when a finalnumber of tiles in said tile section equals said predefined number oftile receiving regions.
 18. The computer-implemented method of claim 17,wherein said tile section comprises a plurality of tiles.
 19. Thecomputer-implemented method of claim 18, wherein one of said pluralityof tiles grows disparately from another of said plurality of tiles.